HSC 2013-14 MX1 Marathon (archive) (1 Viewer)

HeroicPandas · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

phoenix159 said:
Integration - Volume of Solids of Revolution

A hemispherical bowl of radius a units is filled with water to a depth of a/2 units. Find, by integration, the volume of the water.

Whats the answer? V = ?

phoenix159 · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

V = 5/24 pi a³

clintmyster · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

Here's a trick one for you guys:

http://img541.imageshack.us/img541/535/wwgo.png?

HeroicPandas · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

phoenix159 said:
V = 5/24 pi a³

Volume spun around y-axis, hence dy

$V = \pi \int_a^b x^2 dy =\pi \int_{-\frac{a}{2}}^a \left ( a^2 - y^2 \right )dy$

Then integrate and sub in values to get the answer

ForOneEon · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

phoenix159 said:
Integration - Volume of Solids of Revolution

A hemispherical bowl of radius a units is filled with water to a depth of a/2 units. Find, by integration, the volume of the water.

I'm going to feel so stupid for this, but here we go! (incorporation of 4u stuff >_<)

$\text{Imagine the bowl has been sliced perpendicular to the water level.}$
$\text{Now, imagine a triangle formed using the radius of the hemisphere (a) as the hypotenuse and the vertical side (a/2).}$
$\text{The remaining side, the radius of the uppermost circle at height (a/2) is} \frac {a \sqrt{3}}{2} \\$
$$Now, Area =$ \int^{\frac {a \sqrt{3}}{2}}_{0} \pi r^2 \, dr$
$= \frac {\pi r^3}{3} $with those limits that I can't write properly$$
$= \frac {\pi a^3 \sqrt {3}}{8}$

Hope it helps.

HeroicPandas · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

clintmyster said:
Here's a trick one for you guys:

http://img541.imageshack.us/img541/535/wwgo.png?

was this in 2u marathon?

ForOneEon · Oct 21, 2013

Re: HSC 2013 3U Marathon Thread

Oh god, I feel stupid for forgetting it was as simple as rotating a circle about the y-axis. *facepalm*

omgiloverice · Oct 22, 2013

Re: HSC 2013 3U Marathon Thread

double post

omgiloverice · Oct 22, 2013

Re: HSC 2013 3U Marathon Thread

A ladder is to be carried down a hallway u units wide. At the end of the hallway there is a right-angled turn into another hallway of the same width. What is the greatest length of the ladder that can be carried horizontally around the corner?

HeroicPandas · Oct 22, 2013

Re: HSC 2013 3U Marathon Thread

omgiloverice said:
A ladder is to be carried down a hallway u units wide. At the end of the hallway there is a right-angled turn into another hallway of the same width. What is the greatest length of the ladder that can be carried horizontally around the corner?

Is it $2\sqrt{u}$ ?

omgiloverice · Oct 22, 2013

Re: HSC 2013 3U Marathon Thread

HeroicPandas said:
Is it $2\sqrt{u}$ ?

Holy crap I have no idea, this question was harder then I initially thought... sy halp plz

HeroicPandas · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

$$Evaluate this limit: $\lim_{x \rightarrow0} \frac{\sin4x}{sinx} $ by expanding the numerator or \underline{otherwise}$

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

HeroicPandas said:
$$Evaluate this limit: $\lim_{x \rightarrow0} \frac{\sin4x}{sinx} $ by expanding the numerator or \underline{otherwise}$

The limit is obviously 4 since the 'sines cancel out'

=p

RealiseNothing · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Carrotsticks said:
The limit is obviously 4 since the 'sines cancel out'

=p

like this?

HeroicPandas said:
$$Evaluate this limit: $\lim_{x \rightarrow0} \frac{\sin4x}{sinx} $ by expanding the numerator or \underline{otherwise}$

$\frac{sin4x}{sinx} = \frac{sin(x+x+x+x)}{sinx} = \frac{sinx+sinx+sinx+sinx}{sinx} = \frac{1+1+1+1}{1}$

$=4$

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Bravo! State rank material right there.

clintmyster · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

HeroicPandas said:
was this in 2u marathon?

not as far as I know

integral95 · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

RealiseNothing said:
like this?

$\frac{sin4x}{sinx} = \frac{sin(x+x+x+x)}{sinx} = \frac{sinx+sinx+sinx+sinx}{sinx} = \frac{1+1+1+1}{1}$

$=4$

Wow GENIUS !!!

HeroicPandas · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

omg...................................

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

Consider two points A and B on a parabola, forming a chord AB that cuts off some 'segment' of the parabola. Let the area of that segment be X.

Now choose a third point C in between A and B such that the tangent is parallel to the chord AB. The points ABC form a triangle with area Y.

Prove that 3X = 4Y.

Carrotsticks · Oct 23, 2013

Re: HSC 2013 3U Marathon Thread

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